approximation in multidimensional pricing
TRANSCRIPT
Approximation in Multi-Dimensional Pricing
Shuchi Chawla∗ Jason D. Hartline† Robert D. Kleinberg‡
Abstract
There are concise characterizations of optimal mechanisms and monopoly pricings in single-dimensionalcases (e.g., Myerson (1981)). In multi-dimensional (e.g., multi-product) settings there are no such char-acterizations. In many simple, relevant settings optimal and even near-optimal item-pricings are com-putationally intractable. We consider item-pricing for a unit-demand consumer with values for eachitem drawn independently, perhaps not identically, from a known distribution. We draw a close analogybetween this revenue maximization problem and Myerson’s single item auction problem. We show thatconsidering the problem in virtual valuation space instead of valuation space simplifies the problem ofapproximating the optimal item-pricing. Indeed, with a constant virtual price for each item a seller canapproximate the revenue of the optimal item-pricing. We prove an approximation factor of three.
1 Introduction
For all its successes in characterizing optimal mechanisms in single-parameter settings, such as single-itemauctions (Myerson, 1981) and exchanges (Myerson and Satterthwaite, 1983), prior work in optimal mech-anism design has failed to identify the salient characteristics of optimal mechanisms for multi-parametersettings. Even in the seemingly simple setting where there is only a single agent with values for outcomesdrawn from a distribution and a mechanism is simply a pricing over outcomes, the optimal mechanism hasonly been described as the solution to a constrained optimization problem given by the incentive compat-ibility constraints, feasibility constraints, and the designer’s objective (e.g., profit maximization). We areunlikely to obtain a crisp characterization as impossibility results from economics literature show that forany mechanism there is a distribution for which it is optimal and recent work in computer science showsthat this constrained optimization problem is NP-complete (Guruswami et al., 2005).1
For contrast in the single-parameter setting, the seminal result of Roger Myerson (1981) that statesthat the optimal single-item auction is Vickrey with an appropriately chosen reservation price (e.g., fori.i.d. distributions satisfying the monotone hazard rate condition). This suggests that when searching thehigh-dimensional space of all mechanisms, one can restrict attention to a single-dimensional subspace inwhich an optimal mechanism is guaranteed to lie (i.e., the subspace of all Vickrey mechanisms parameterizedby a reservation price). Even in the more general case of independent but not necessarily identical agentvaluations, Myerson’s result conceptually simplifies the optimization problem by first suggesting transformingthe valuation space into virtual valuation space and then performing the trivial optimization of selecting theagent with the highest positive virtual valuation to win the item. Unfortunately, many of the insights of thissingle-parameter problem fail to carry over to the multi-parameter setting (see, for example, McAfee andMcMillan (1988) and Jehiel et al. (1999)).
In this paper we focus on a multi-item unit-demand pricing problem. We assume a seller faces a singleunit-demand agent with valuations for each item distributed independently but not necessarily identically.Our seller will post a take-it-or-leave-it pricing on items. Given the prices of items, the agent then selects
∗Computer Sciences Department, University of Wisconsin - Madison. Email: [email protected].†Electrical Engineering and Computer Science, Northwestern University. Email: [email protected].‡Dept. of Computer Science, Cornell University. Email: [email protected] NP-completeness means that under standard computational assumptions there is no description of the optimization
problem that would enable a computer to find an optimal solution in any reasonable amount of time.
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the item that maximizes her utility, the difference between her value and the price posted. Our problem isto determine the seller’s pricing on items (a.k.a., item-pricing) that maximizes her revenue. Naturally, thisis a multi-dimensional optimization problem.
While we consider only a single agent and item-pricings, this problem has important connections tomechanism design. Notice that in classical optimal mechanism design theory, the multi-agent single-itemoptimal auction actually uses as a black box the single-agent single-item optimal mechanism: the optimalreserve price in the Vickrey auction is precisely to the optimal price to post to a single agent. This is aconnection we expect to continue to be interesting in multi-dimensional settings. One example of this iswork in computer science where a certain multi-item mechanism design problem is approximately reducedto the multi-item pricing problem (Balcan et al., 2005).
We approach the problem of characterizing good multi-parameter pricings through the lens of approxima-tion. We identify a small class of pricings, optimizable over a single-dimensional search space, with a simpleintuitive description and the property that for any distribution there is a mechanism in this class that isapproximately optimal, that is, it obtains revenue that is a large fraction of the optimal revenue. Propertiesof this class of mechanisms, then, provide insight into the nature of optimal multi-parameter pricings andmechanisms. Furthermore, our characterization implies an efficient procedure for finding the approximatelyoptimal pricing, whereas the optimal pricing cannot be found efficiently even using a computer. In thissense, since we expect sellers to only be able to perform simple optimizations, our approximation is morepractically relevant than an optimal solution.
Our main result states that for the multi-item unit-demand pricing problem, a single price in virtualvaluation space for all items is approximately optimal (for general product distributions). This statementconcisely captures sufficient conditions for approximate optimality. It reduces the optimization problemfrom a multi-dimensional problem to a single-dimensional problem (namely that of finding the best single“virtual” price). It also yields a computationally tractable procedure for finding the approximately optimalpricing. In conclusion, by backing off of the impossible problem of understanding the exact optimal pricingin multi-parameter settings we have arrived at an approximation result with all of the desirable propertiesof its single-parameter optimal counterpart, except for exact optimality.
We point out that even in the i.i.d. case (that is when all item values are identically distributed) the multi-item pricing problem is non-trivial. If one restricts attention to single-price item-pricings (those where eachitem is priced at the same amount) the optimization problem simplifies: just consider the distribution of themaximum valuation (maxi vi) and solve this problem as a single-consumer single-item revenue maximizationproblem. Unfortunately, the optimal item pricing in this symmetric setting is not necessarily a single-priceitem-pricing: consider two items, each with a value independently equal to 1 with probability 2/3 and 2 withprobability 1/3; then a simple calculation shows that the pricings (1, 2) and (2, 1) are optimal with respectto revenue2 and the pricings (1, 1) and (2, 2) are strictly sub-optimal.
Methodology
We follow a standard approach to approximation. As the optimal performance is a difficult quantity tounderstand (e.g., from lack of concise description of the optimal pricing) we instead look for an upper boundon the optimal performance. In our setting this upper bound comes from an analogy between our single-agentmulti-item pricing problem and the multi-agent single-item auction problem. Notice that the distributionalstructure of these two problems is the same. We show that the performance optimal multi-agent single-itemauction, which is well understood (Myerson, 1981), gives an upper bound on the optimal performance of asingle-agent multi-item pricing.
Consider the i.i.d. case. Here the optimal single-item auction is Vickrey with a reserve price. Whencomparing this to a multi-item pricing with a single-price across all items, the main distinction is in thepayment rule in the event that more than one item value is above the posted price. In the single-itemauction, competition results in a higher sale price: the winner pays the second highest value which is at least
2Here we have assumed that whenever the consumer faces a tie, i.e. two or more items bring equal utility to her, the sellermay decide the outcome (in particular, in favor of the most expensive item). The seller can enforce this by giving a negligiblysmall discount to the consumer for the most expensive item.
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the reserve. In the multi-item pricing, the agent simply chooses the item with maximum value and the saleprice remains the price posted. We show that the competition in the single-item auction can be simulatedin the multi-item pricing setting by setting a somewhat higher single price for all items such that there is aconstant probability that exactly one item value exceeds it. To complete this last step of the approximationresult, we show that the performance of this pricing is within a constant factor of the performance of theoptimal single-item auction.
This approach extends to non-identically distributed distributions as well. Once again we start with thereserve prices of the optimal single-item auction and raise them appropriately to account for the lack ofcompetition in the pricing setting. We show that the right rate at which to increase the prices of each itemis constant in virtual valuation space: competition can be simulated by setting a single-virtual-price for allitems with the property that there is a constant probability of exactly one virtual value exceeding it.
Results
For i.i.d. distributions we identify a single-price posted across all items that gives a 2.17-approximation to theoptimal item-pricing, that is, it obtains at least a 1/2.17 fraction of the optimal revenue). For independentnon-identically distributed distributions a single-price cannot approximate the optimal item-pricing (implyingthat there is not an approximate Bulow-Klemperer analog for multi-dimensional settings; Section 3.3).Instead we identify a single-virtual-price posted across all items that gives a 3-approximation to the optimalitem-pricing. Section 3 contains our basic arguments for the case when all the input distributions are regular(see Definition 4). We extend our technique to the non-regular case (yielding the same result) in Section 4.These approximations are computationally tractable: we give an algorithm that can be implemented on acomputer to quickly determine approximately optimal prices (see Section 5).
Related Work
Multi-dimensional optimal mechanism design has received much attention in economics literature over thelast three decades. Some of this work (see, for example, McAfee and McMillan (1988) and Jehiel et al.(1999)) has focused on developing simple characterizations of incentive constraints in the multi-dimensionalsetting. These can then be used to develop algorithms for solving the optimization problem in special casessuch as when the so-called “single-crossing” condition is satisfied. Unfortunately, these approaches have notlead to a general purpose algorithm for finding the optimal mechanism.
Literature on multi-dimensional pricing has also focused primarily on special cases where simple charac-terizations of the optimal solution exist, and the optimization problem can be solved exactly. For example,Armstrong (1996) gives an algorithm for finding the optimal pricing when a strong condition on the con-sumers’ value distribution is satisfied. McAfee and McMillan (1988) provide conditions under which theoptimal selling mechanism is a price schedule, and Manelli and Vincent (2006) follow up with conditions un-der which the best price schedule must be a deterministic one. Rochet and Chone (1998) identify conditionson the value distribution under which their “sweeping” approach finds the optimal solution. In contrast wedevelop a general purpose algorithm for finding approximately-optimal solutions that works under very mildassumptions on the consumers’ values – the consumers are unit-demand and values for different goods areindependent.
To our knowledge the only work in economics that considers approximately-optimal solutions to thepricing problem is that of Armstrong (1999). Armstrong assumes that consumers’ utility is separable acrossproducts, and under this assumption using “large number” arguments finds a solution that is asymptoticallyoptimal. The separability condition essentially breaks the problem into multiple single-dimensional problems,each corresponding to a single item, whereas in our model with unit demands, prices for each item grealyeffect the demand for others.
Our work is based heavily on the foundational economics work in single-dimensional optimal mechanismdesign of Myerson (1981), Riley and Samuelson (1981), and Bulow and Roberts (1989).
Single and multi-agent pricing problems, in particular the computational (in)tractability of their solution,have recently been under intense scrutiny by computer science literature on algorithmic pricing (Aggarwal
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et al., 2004; Balcan and Blum, 2006; Briest and Krysta, 2007; Guruswami et al., 2005; Hartline and Koltun,2005). This prior work is motivated in part by the close connection these problems have to mechanismdesign, especially in prior-free settings, e.g., (Balcan et al., 2005; Guruswami et al., 2005; Aggarwal andHartline, 2006). The algorithmic pricing literature contains only a few positive results on approximation,e.g., (Balcan and Blum, 2006; Hartline and Koltun, 2005); otherwise, it has been largely negative showingthat good approximation are computationally intractable (Briest and Krysta, 2007; Briest, 2008; Demaineet al., 2006). This motivates the search for relevant special cases where algorithmic theory gives an improvedunderstanding of pricing. Our work can be seen as a particularly relevant such special case where goodapproximations are tractable.
2 Notation and preliminaries
This paper builds a concrete relationship between the following two problems.
Definition 1 (Bayesian Single-item Auction Problem (BSAP))Given,• a single item for sale,• n consumers, and• distribution F from which the consumers’ valuations for the item are drawn.
Goal: the seller optimal auction for F.
Definition 2 (Bayesian Unit-demand Pricing Problem (BUPP))Given,• a single unit-demand consumer,• n items for sale, and• distribution F from which the consumer’s valuations for each item are drawn.
Goal: the seller optimal item-pricing for F.
The input to each problem is a valuation profile that is drawn from a distribution over n-tuples ofvaluations. We use v = (v1, . . . , vn) for the random variable representing the valuation vector as well as itsinstantiation. In BUPP we interpret vi as the valuation of the (single) consumer for item i, while in BSAPwe interpret vi as the valuation of consumer i for the (single) item. The value vi is drawn independentlyfrom the distribution Fi over the range [ℓi, hi]. Following standard notation, we use v−i to denote all thevaluations except the ith one. F = F1 × · · · × Fn denotes the product distribution from which v is drawn,and fi(vi) denotes the probability density of valuation vi. (Unless otherwise specified, we will assume thatthe distribution of vi is absolutely continuous with respect to Lebesgue measure, so the density functionfi(vi) is well-defined.)
We focus on the special case where F is the product distribution F1×· · ·×Fn. In this setting BSAP wassolved precisely by Myerson (1981). Our aim is to approximately solve BUPP.
Regularity and the Monotone Hazard Rate condition
In much of the paper we will assume that the distributions Fi satisfy the so-called regularity conditiondefined below. This condition is satisfied by any distribution that has a monotone hazard rate (MHR) andis a standard assumption. In the single-consumer, single-item case, this condition essentially implies thatthe revenue as a function of price is concave and therefore has a unique maximum. We note that uniform,normal, and exponential distributions all have monotone hazard rate.
Definition 3 (Monotone Hazard Rate) A one-dimensional distribution F with density f is said to sat-
isfy the monotone hazard rate (MHR) condition if the “hazard rate” of the distribution, f(v)1−F (v) , is a mono-
tonically non-decreasing function of v.
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Definition 4 (Regularity) A one-dimensional distribution F with density f is said to be regular (or satisfy
regularity) if v − 1−F (v)f(v) is monotonically non-decreasing for all v.
When each of the Fis satisfy regularity, we say that the product distribution F = F1×· · ·×Fn is regular.In Section 4 we extend our results to distributions that do not satisfy regularity. This is also called thenon-regular case in the literature.
Virtual valuations and the Bayesian Single-item Auction Problem (BSAP)
The Bayesian Single-item Auction Problem (BSAP) is described as follows: there is a single item for sale andn bidders with values given by the vector v; each bidder’s value vi is drawn independently from a distributionFi; the goal of the mechanism designer is to design an incentive-compatible auction so as to maximize therevenue obtained by the seller from the sale of the item.
In one of the seminal works of Bayesian mechanism design, Myerson developed a mechanism for thisproblem that obtains the maximum revenue for the seller over the class of all incentive-compatible mecha-nisms (Myerson, 1981). Myerson’s mechanism (denoted M hereafter) is based on the following definitions:
Definition 5 The virtual valuation of bidder i with valuation vi drawn from Fi is
φi(vi) = vi −1− Fi(vi)
fi(vi). (1)
The virtual surplus of a single-item auction is the virtual valuation of the winner.
Note that for any regular distribution, the virtual valuation φ(v) is a non-decreasing function of thevaluation v.
Theorem 1 (Myerson (1981)) Any incentive-compatible auction A has expected revenue equal to its ex-pected virtual surplus.
The virtual valuation of a bidder essentially denotes the marginal revenue obtained by allocating the itemto this bidder. A simple consequence of Myerson’s Theorem is that maximizing revenue (in expectation) isequivalent to maximizing virtual surplus; therefore, for regular distributions the optimal single-item auctionis the one that sells to the bidder with the highest non-negative virtual valuation3. Incentive compatibilityconstraints then imply that the payment of the optimal mechanism should be the value at which the winningbidder’s virtual valuation equals the second highest virtual valuation, i.e., the payment is equal to thevirtual-valuation-inverse of the second highest virtual valuation. (If all other virtual valuations are negative,we consider the second highest virtual valuation to be 0.) It is sometimes convenient to view Myerson’smechanism as offering each bidder a take-it-or-leave-it price, where the price offered to bidder i is equalto φ−1
i (νi) and νi = maxj 6=i max(φj(vj), 0). Notice that only the bidder with the highest virtual valuationwould accept such a take-it-or-leave-it offer.
We use RA to denote the expected revenue of an incentive-compatible auction A for BSAP. RM denotesthe expected revenue of Myerson’s auctionM.
Corollary 2 When F satisfies regularity, RM ≥ RA for all incentive-compatible auctions A.
We can also apply Myerson’s Theorem to a variant of the single-item auction problem where the sellerhas some reservation value, ν, for keeping the item. The virtual surplus in this setting would be the virtualvaluation of the winning bidder or ν if the item remains unsold. Myerson’s Theorem says that the optimalmechanism (by maximizing virtual surplus) sells the item to the bidder with the highest virtual valuation ifand only if that virtual valuation is at least ν. We denote this mechanism byMν .
In other words, if we use the notation χ(A) to denote the probability that the item is unsold when usinga given auction mechanism A, then for every incentive compatible mechanism A we have:
3In the non-regular case, that is when Definition 4 is not satisfied, this auction may not be incentive compatible. In Section 4we describe the modifications required to obtain an optimal incentive compatible auction in that case.
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Corollary 3 When F satisfies regularity, for all incentive-compatible auctions A,
RMν + ν · χ(Mν) ≥ RA + ν · χ(A).
The observation of the following fact completes our preliminary discussion of virtual valuations andoptimal auctions.
Fact 4 Virtual valuations satisfy φi(vi) ≤ vi.
The Bayesian Unit-demand Pricing Problem (BUPP)
The Bayesian Unit-demand Pricing Problem (BUPP) is described as follows: there are n items for sale anda single consumer with unit-demand, quasi-linear preferences given by the vector v; the consumer’s value vi
for item i is drawn independently from a distribution Fi; the goal of our algorithm is to determine a pricevector p = (p1, · · · , pn) such that the expected revenue Rp, as defined below, is maximized.
Rp =∑
ipi ·Prv∼F [(vi − pi) = maxj≤n(vj − pj) ∧ vi − pi ≥ 0]
The connection between BUPP and BSAP
The connection between BUPP and BSAP starts with the case where n = 1 and the two problems areidentical. Here the optimal pricing (and the optimal auction) is to offer the item at a virtual price of zero.
For larger n, the competition between bidders in BSAP will allow the optimal auction, RM, to obtainat least the revenue, Rp, of any pricing p for BUPP.
Lemma 5 For any price vector p, RM ≥ Rp.
Proof: For a given pricing p, consider the following mechanism Ap: given a valuation vector v, weallocate the item to the bidder i with vi ≥ pi that maximizes vi − pi. Prices are determined by the standard“threshold payment” rule: the winning bidder, i, pays the minimum bid value which would still make i thewinner. Ap is incentive compatible because it gives a monotone allocation procedure: if a winning bidderunilaterally increases her bid, she still wins. Therefore, RAp ≤ RM.
Now consider any valuation vector v and suppose that Ap allocates the item to bidder i. Then theminimum bid at which this bidder is allocated the item is pi + maxj 6=i(vj − pj , 0), which is at least pi.Therefore, the revenue of Ap when the valuation vector is v is at least pi. However, the revenue of thepricing p when the valuation vector is v is exactly pi. Therefore, RAp ≥ Rp. Combining the two inequalitiesproves the lemma.
Our motivation to relate BUPP to Myerson’s solution of BSAP stems from the observation that as thenumber of bidders gets large (especially in the case of identically distributed valuations), the price offered toa bidder in Myerson’s mechanism as a function of other agents’ bids becomes tightly concentrated arounda single value (the expectation of the virtual-value-inverse of the maximum over virtual valuations of otherbidders). This value is a reasonable candidate for the price of item i in the pricing problem, and is indeedroughly what we use (with some modifications to allow for a simpler analysis). Thus, in examining the actualoutcomes of the optimal auction, we gain intuition for outcomes our pricing should attempt to mimic.
This approach has obstacles that must be overcome. Myerson’s mechanism, by allowing the price to be anappropriate function of other bidders’ values, ensures that only one bidder accepts the offered price; whereasin BUPP, with some probability, more than one of the values is above its corresponding offer price. Whenthis happens the consumer gets to pick which item to buy. In this case the price earned by our solution toBUPP may be much worse than the price earned by Myerson’s solution to BSAP for the same valuationvector. We can get around this problem by making use of regularity (e.g., by monotone hazard rate) andother techniques from optimal mechanism design when regularity does not hold.
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Virtual Pricings and Auctions
We will be searching over price space for an item pricing p with revenueRp that approximates the revenue ofMyerson’s optimal auction, RM. Our search will focus on the subspace of “constant-virtual-price” pricings,that is, pricings that are equal in each coordinate in virtual valuation space. Central to our discussion willbe correspondences between constant virtual pricings and optimal auctions. We define below the necessaryterminology for discussing the regular case, i.e., when the virtual valuation functions are monotone non-decreasing. Notice that we will allow virtual valuation functions to be constant over subranges. Such virtualvaluations functions result in a multiplicity of pricings of constant virtual price as well as a multiplicity ofoptimal auctions.
The following definitions allow us to describe constant-virtual-price pricings for regular distributions:
• P is the set of constant-virtual-price pricings, i.e., P = p : ∃ν ∀i φi(pi) = ν
• The virtual-price of a constant-virtual-price pricing p ∈ P is φ(p) = φi(pi) for all i.
• Pφ=ν is the set of pricings with constant virtual price ν, i.e., Pφ=ν = p : p ∈ P ∧φ(p) = ν. Whereit is clear from the context, we use shorthand notation Pν .
Note that in the BSAP, when the seller has a reservation value ν for keeping the item and the highestvirtual valuation is equal to ν, the seller is indifferent towards selling or not selling the item in the optimalauction. Indeed different ways of breaking the tie between selling and not selling give rise to multiple differentoptimal auctions. The following definition allows us to refer to optimal auctions with particular tie-breakingrules. Notice that we still leave ambiguous the tie-breaking rule for selecting between two agents with thesame virtual value.
• Mp is the auction that sells to the agent i with vi ≥ pi and the highest φi(vi) among such agents. Ifno such agent exists the item remains unsold. When p ∈ Pν , Mp is an optimal auction for a sellerwith reservation value ν. When we do not want to be specific about the tie-breaking rule, we will usethe predefined short-hand notationMν to represent any of the optimal auctions.
The following definitions allow us to describe the probability that no item is allocated by an auction orpricing.
• χ(A) is the probability no item is sold at by auction A. χ(p) is the probability no item is sold atpricing p. Notice that χ(Mp) = χ(p).
• (For implicit p) qi is the probability that vi ≥ pi when vi ∼ Fi. In other words, qi = 1 − Fi(pi).Clearly, χ(p) = Pr[vi < pi for all i] =
∏
i Fi(pi) =∏
i(1− qi).
• Pχ=x is the set of constant-virtual-price pricings with probability of no sale equal to x, i.e., Pχ=x =p : p ∈ P ∧ χ(p) = x. Where it is clear from the context, we use shorthand notation Px.
Finally we define the appropriate notation to relate the virtual price of constant-virtual-price pricings tothe probability that no item is allocated. Among the prices, p ∈ Pν , with a constant virtual price ν, thecoordinate-wise smallest one has the lowest probability that no item is sold and the coordinate-wise highestone has the highest probabiltiy that no item is sold. Any probability in between can be attained at somepricing in the set. Any pricing p′ ∈ Pν′ with constant virtual price ν′ strictly higher (resp. lower) than νhas a strictly higher (resp. lower) probability of no-sale than any pricing with constant virtual price ν.
• νx = φ(Px) is the virtual-price φ(p) of all pricings p ∈ Px. Notice that by the above discussion φ(p)are equal for all p ∈ Px.
• Xν is the set of probabilities of no-sale for constant-virtual-price ν pricings, i.e., Xν = x : νx = ν.
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The computational model
Much of this paper focuses on an analysis that reduces the multi-dimensional price optimization problem toa single-dimensional uniform virtual price optimization. To actually compute (or approximate) the pricingthat our analysis suggests, we consider two different computational models:
• (Discrete explicit) In this model, each of the distributions Fi is a discrete distribution with smallsupport. These distributions are specified explicitly, and our algorithm is required to run in timepolynomial in the number of items n, and the size of the largest support.
• (Continuous with oracles) In this model, the distributions Fi are continuous with known supports[ℓi, hi]. The algorithm is provided the following oracles: an oracle to determine Fi(v) given a value vand an index i, an oracle to determine the density fi(v) given a value v and an index i, an oracle tosample from the product distribution F, and finally an oracle giving some p ∈ Pφ=0.
4 The algorithmis required to run in time polynomial in the number of items n and the range (maxi hi)/(mini ℓi).
3 Approximating pricing in the regular case
In this section we demonstrate that the unit-demand optimal pricing that uses a single virtual price for allitems obtains a constant fraction of the revenue of the optimal single-item auction (and also of the revenueof the optimal pricing). Specifically we obtain a 3-approximation for general (regular) product distributions,and an improved 2.17-approximation when all the valuations are distributed identically.
3.1 Analysis
In this section we show that either every p ∈ Pχ=1/2 or every p ∈ Pφ=0 satisfies
Rp ≥ RM/3.
That is, a single price in virtual valuation space approximates the optimal auction (and thus, by Lemma 5,the optimal pricing). We first show that RM can be bounded from above in terms of ν and RMν (below,Corollary 6). We then bound from above both ν (below, Lemma 7) and RMν (below, Lemmas 8, 9,Corollaries 10, 11) by appropriate multiples of Rp.
Corollary 6 For any ν ≥ 0, RMν + ν · χ(Mν) ≥ RM.
Proof: This result follows by invoking Corollary 3 with A = M and noting that χ(A) ≥ 0 for everymechanism A.
Lemma 7 For p ∈ P, Rp ≥ (1− χ(p)) · φ(p).
Proof: Fact 4 implies that pi ≥ φi(pi) = φ(p) for all i. By definition, 1 − χ(p) is the probability that anitem is sold. Thus, Rp ≥ (1 − χ(p)) · φ(p).
Lemma 8 For any p, Rp ≥ χ(p) ·∑
i piqi.
Proof: The revenue Rp is bounded below by the summation, over all i, of pi times the probability that i isthe unique index satisfying vi ≥ pi, i.e.
Rp ≥∑
ipi ·(
qi
∏
j 6=i(1− qj)
)
=∑
ipiqi
χ(p)1−qi
≥ χ(p)∑
ipiqi.
4Our techniques can be used to remove the assumption that some p ∈ Pφ=0 is given. However, it is natural to assume thatthis is known as each pi is the solution to the “solved” problem of optimally pricing a single item with value distribution Fi.
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Before we bound RMν in terms of Rp we need a new definition. For a given auction A and prices p,consider the event, EA,p, in which the winner, say i, has vi ≥ pi. Let RAp be the contribution to the expected
revenue of A from such events. That is, RAp is the expected revenue of A conditioned on EA,p times the
probability that EA,p happens. The following lemma shows that RAp can be bounded in terms of Rp.
Lemma 9 Under regularity, for any p with φi(pi) ≥ 0 for all i and any incentive-compatible auction, A,we have RAp ≤
∑
i piqi.
Proof: We modify A to get A′ with RAp ≤ RA′
p = RA′
. If A sells to a bidder i with valuation at leastpi, A
′ does the same; otherwise, A′ does not sell the item5. It is easy to see that A′ is incentive compatibleif A is. Since the modified auction never sells to a bidder i with vi < pi, it is immediate that RA
′
p = RA′
.Moreover, if the winner i of A has value vi ≥ pi, this modification does not change the allocation and onlyincreases the payments. So RAp ≤ R
A′
p . Now we get an upper bound on RA′
.A′ is an auction that never sells to a bidder i at price less than pi. Certainly, its revenue is less than the
optimal auction satisfying the same price constraint. Moreover, this optimal auction’s revenue is less thanthe optimal auction that does not have a supply constraint (i.e., it can sell multiple copies of the item). Wenow show that the optimal auction, that can potentially sell an item to all bidders at once, but is constrainedto use prices at least p, has revenue precisely
∑
i piqi.Since it can sell to all bidders, an optimal auction would make its allocation decisions for each bidder
independently. Consider bidder i. If vi ≥ pi then i has non-negative virtual surplus. Regularity andMyerson’s theorem (expected revenue equals expected virtual surplus) imply that this optimal auction wouldsell to bidder i. Of course when vi < pi the auction cannot sell to bidder i. The auction that sells to i if andonly if vi ≥ pi has expected payment precisely piqi. Summing over all bidders, the total expected revenue ofthis optimal auction would be
∑
i piqi.
We can now put lemmas 8 and 9 together to get the following:
Corollary 10 Under regularity, any auction A and any p with φi(pi) ≥ 0 for all i satisfies
χ(p) · RAp ≤ Rp.
Since RMp = RMp
p we have:
Corollary 11 Under regularity, any p with φi(pi) ≥ 0 for all i satisfies χ(p) · RMp ≤ Rp.
We now state and prove our main lemma.
Lemma 12 Under regularity, if ν1/2 ≥ 0 then p ∈ Pχ=1/2 satisfies RM ≤ 3Rp.
Proof: Recall that χ(Mp) = χ(p) = 1/2.
RM ≤ RMp + φ(p)χ(p) Corollary 6
≤ 1χ(p)R
p + φ(p)χ(p) Corollary 11
≤ 1χ(p)R
p + χ(p)1−χ(p)R
p Lemma 7
= 3Rp Using x = 1/2.
Notice that χ(p) = 1/2 is indeed the optimal choice for χ above.
The only loose end to wrap up now is the case that ν1/2 < 0. In this case, we can show that p = r(0) isa 2-approximation using only Corollary 10.
5Notice that incentive compatibility constraints require that the payment in A′ when allocating to bidder i is maximum ofthe payment i would have made in A and pi.
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Lemma 13 Under regularity, if ν1/2 ≤ 0 then p ∈ Pφ=0 satisfies RM ≤ 2Rp.
Proof: Note that for all p ∈ Pφ=0, RMp is RM and χ(p) > 1/2. Applying Corollary 11 with p we can
conclude that RM ≤ 2Rp.
We combine Lemmas 5, 12, and 13 to get the main theorem of the paper.
Theorem 14 Under regularity either every p ∈ Pχ=1/2 or every p ∈ Pφ=0 is a 3-approximation to theoptimal pricing, the former if ν1/2 ≥ 0 and the latter if ν1/2 ≤ 0.
3.2 Analysis of the i.i.d. case
We now consider the i.i.d. case where all valuations are distributed according to F , with density functionf , and virtual valuation function φ(·). In this case, for any x (respectively ν), there exist pricings in Px
(resp. Pν) that are single valued, that is the price for every item is the same. Constrained to this classof “single-price” pricings, it is easy to analytically describe the optimal pricing and even easier to computeit via a sampling algorithm. Let Fmax be the distribution for maxi vi when vi is distributed from Fi. Theoptimal single price to use is the p that maximizes p · (1− Fmax(p)).
Recall that we showed in the introduction that a single-price pricing is not necessarily optimal. In thissection we show that such a pricing is in fact fairly close to an optimal pricing in the i.i.d. case. For asingle-price pricing p = (p, . . . , p) that is implicit, we let q be the probability that a valuation drawn from Fis at least p. Let p1/e denote the single-price pricing for which q = 1/n. We will abuse notation to let ν1/e
be the corresponding virtual valuation, i.e., ν1/e = φ(p1/e). As before, our final solution will either be thepricing p1/e or any pricing p ∈ Pφ=0.
We show that one of these pricings is a 2.17-approximation to the optimal pricing. Our analysis is neartight—towards the end of this section we give an example for which no single-price pricing is better thana (2 − o(1))-approximation to the optimal pricing; Therefore, to beat the factor of 2, one must necessarilyconsider non-single-price pricings.
The motivation for our abuse of notation is the following lemma.
Lemma 15 For distribution F and pricing p = (p, . . . , p) such that 1 − F (p) = 1/n, we have χ(p) ≤ 1/e,and this bound is asymptotically tight.
Proof: By definition, χ(p) = (F (p))n = (1− 1/n)n ≤ 1/e.
The following adaptations of results from the preceding section allow us to prove the main result of thissection. The improvement we obtain in the i.i.d. case, over the general case, comes from the following. First,Fact 16 gives a slightly better lower bound on Rp than Lemma 7. Second, we can use this bound (insteadof Lemma 8) with Lemma 9, to obtain an improved upper bound on RMν in terms of Rp when ν1/e ≥ 0and χ(p) < 1/e.
Fact 16 Rp = p(1− χ(p)) for any p = (p, . . . , p).
Lemma 17 For a regular and i.i.d. distribution, with p = F−1(1 − 1/n), p1/e = (p, . . . , p), and ν1/e =φ(p1/e), ν1/e ≥ 0 implies RM ≤ 2.17 Rp1/e .
Proof: Notice that the probability that no item is sold in Mν1/eand under pricing p1/e is the same, i.e.,
x = χ(Mν1/e) = χ(p1/e) ≤ 1/e.
RM ≤ RMν1/e + ν1/e · x Lemma 6
≤∑
ipq + ν1/e · x Lemma 9
≤ p + ν1/e · x Since q = 1/n
≤ (1 + x) · p Since ν1/e ≤ p
= 1+x1−xR
p1/e Fact 16.
10
Finally, for any 0 < x ≤ 1/e, (1 + x)/(1− x) is at most 2.17.
We now need to handle the case where ν1/e < 0 by showing that p ∈ Pφ=0 is good. Unfortunately, weonly know that χ(p) is approximately at least 1/e whereas Lemma 13 needed χ(p) ≥ 1/2. What is neededis a tighter lower bound on Rp in terms of
∑
i piqi = npq than is given by Lemma 8.
Lemma 18 For an i.i.d. distribution, any pricing p = (p, . . . , p) with q = 1 − F (p) ≤ 1/n satisfies Rp ≥npq/2.
Proof: For the pricing p, the probability that a sale is made is exactly
Pr[∃i | vi ≥ p] = 1− (1− q)n.
Using Taylor’s expansion and q ≤ 1/n we can simplify this expression as follows: (1−q)n < 1−qn+ 12q2n2 ≤
1− 12qn. Therefore the expected revenue of p is Rp = p(1− (1− q)n) ≥ npq/2.
Lemma 19 For a regular and i.i.d. distribution with ν1/e ≤ 0, any pricing p ∈ Pφ=0 satisfies RM ≤ 2Rp.
Proof: First, q is at most 1/n so Lemma 18 implies that Rp ≥ npq/2. Myerson’s mechanism never sells ata price less than p = φ−1(0) so Lemma 9 implies that RM = RMp ≤ npq. Thus, Rp ≥ RM/2.
We combine Lemmas 5, 19, and 17 to get the main theorem of this section.
Theorem 20 For a regular and i.i.d. distribution, with p = F−1(1 − 1/n) and p1/e = (p, . . . , p), either thepricing p1/e or every pricing p ∈ Pφ=0 is a 2.17-approximation to the optimal pricing.
The following lower bound shows that this theorem is nearly tight for single-price pricings p = (p, . . . , p).
Lemma 21 There exists an i.i.d. distribution for which no single-price pricing is better than a (2 − o(1))-approximation to the optimal pricing.
Proof: Consider the i.i.d. distribution given by,
Pr[vi = n] = 1n2
Pr[vi = 1] = 1− 1n2 .
An optimal pricing would be to set p1∗ = 1 and pi
∗ = n for all i > 1. This achieves a revenue of 2 − o(1).However, for every pricing p = (p, . . . , p) has Rp ≤ 1.6
3.3 Single-price pricings for general distributions.
Given the simplicity of our solution for i.i.d. distributions, it is natural to ask how well single-price pricingsperform for general distributions. The example below shows that for general distributions single-price pric-ings cannot approximate the optimal pricing to within a constant factor, necessitating a more complicatedsolution, such as the one we give in Section 3.1.
Lemma 22 There exists a distribution F for which no single-price pricing is a o(log n)-approximation tothe optimal pricing.
Proof: Consider the following distribution Fi for vi defined as:
Pr[
vi = ni
]
= 1n
Pr [vi = 1] = 1− 1n .
6In this example the revenue of Myerson’s auction is nearly equal to 2, so the example does not prove any separation betweenthe revenue of Myerson’s auction and that of the optimal pricing.
11
The pricing p which sets pi = n/i achieves a revenue of Ω(log n) because
Rp =
n∑
i=1
(n
i
)
Pr[
vi =n
iand ∀ j < i vj = 1
]
=
n∑
i=1
(n
i
)
·
(
1
n
)
·
(
1−1
n
)i−1
>n∑
i=1
(n
i
)
·
(
1
n
)
·
(
1
4
)
= Hn/4,
where Hn denotes the n-th harmonic number,∑n
i=11i .
On the other hand, if p is any pricing which sets pi = p for some fixed value of p > 1, then
Rp ≤
n∑
i=1
p ·Pr[vi ≥ p]
=∑
1≤i≤n/p
p ·
(
1
n
)
+∑
i>n/p
p · 0
≤ 1.
Thus, no single-price pricing can o(log n)-approximation to the revenue of the optimal pricing.
We now observe that yet another intuition from single-dimensional mechanism design fails to carry overto multi-dimensional settings. Bulow and Klemperer (1996) show that a (profit maximizing) seller of a singleitem to agents with i.i.d. distributed values would prefer to recruit an additional agent and run an efficient(e.g., Vickrey’s) auction than to run the optimal (e.g., Myerson’s) auction. This holds even when there isonly a single agent where the seller would rather run a Vickrey auction on two agents from the distributionthan offer the monopoly price to a single agent.
For our multi-dimensional setting, we might wonder if a seller would prefer to run an efficient auction(e.g., the Vickrey-Clarke-Groves (VCG) mechanism) on two agents with values drawn independently fromthe same distribution or offer a single agent the optimal item-pricing. It is not always the case that the sellerwould prefer the two agent setting, and by a wide margin.
Corollary 23 There exists a distribution F for which the the revenue of the optimal item pricing to a singleagent is Θ(log n) times more than the revenue of VCG on two such agents.
Notice that in our unit-demand setting, VCG on two agents simply awards the agent with the highestoverall valuation their most preferred item. The price this agent must pay is the other agent’s highest overallvaluation. From each agent’s perspective, this is equivalent to being offered a single-price for all items thatis equal to the other agent’s highest valuation. Thus, VCG’s revenue is upper-bounded by twice the revenueof the optimal single-price item-pricing. Of course, Lemma 22 shows that this revenue can be a logarithmicfactor worse than the optimal (multi-price) item-pricing (to a single agent).
4 The non-regular case
In our analysis in Section 3.1, we used regularity (e.g., by the monotone hazard rate assumption) to implythat the functions φi(vi) are non-decreasing, which in turn allowed us to bound the revenue RAp for anauction A in Lemma 9. For BSAP, when regularity does not hold, Myerson applies a fix to the problemby “ironing” the virtual valuation function to make it a non-decreasing function of vi. In this section weshow that by picking a pricing based on ironed virtual valuations instead of the actual virtual valuations,we achieve exactly the same guarantee as in the regular case—the revenue of our pricing is within a factor
12
of 3 of the revenue of Myerson’s mechanism. This result requires new ideas in addition to our approach inSection 3. The main issue we need to deal with is that ironed virtual valuations do not have unique inverses,and we need to pick inverses carefully in order to avoid worsening the approximation factor7.
We briefly describe the ironing procedure below. The reader is referred to Myerson’s paper (Myerson,1981) and a survey of Bulow and Roberts (1989) for more details.
= F(v)
(0,0)
(0,0)
(v) (v)
1
1
R( )
R( )
Figure 1: Converting a virtual valuation function φ to R and φ
The ironing procedure
The ironed virtual valuation function is defined as follows. Consider a single bidder with value v distributedaccording to function F . We assume that the density function f(v) is non-zero for all v ∈ [ℓ, h]. For α ∈ [0, 1],let R(α) denote the revenue generated from offering the item to this bidder at price F−1(α):8
R(α) = F−1(α)(1 − α) =
∫ h
F−1(α)
φ(t)f(t)dt
Let R(α) be the least-valued concave function on [0, 1] with R(α) ≥ R(α) for all α in that range (seeFigure 1). Since R is concave, it is differentiable on a dense subset of [ℓ, h] (cf. Rockafellar, 1970, Theorem25.5). Let r(α) denote the derivative of R wherever defined. The ironed virtual valuation function is definedas below wherever r is defined, and is extended to the full range of v by right continuity.
φ(v) = −r(F (v))
Note that since R(α) is concave and F (v) is non-decreasing, φ(v) is a non-decreasing function. Furthermore,observing that R(1) = R(1) = 0, we get the following:
∫ t=h
t=v
φ(t)f(t)dt = −
∫ t=1
t=F (v)
r(t)dt = R(F (v)) (2)
7A previous version of this paper contained a worse 4-approximation in the non-regular case.8Note that F−1(α) is well-defined because F is a strictly increasing function.
13
Theorem 24 (Myerson (1981)) The expected revenue of any incentive-compatible auction A for BSAPis no more than its expected ironed virtual surplus. Furthermore, if for all i, A has constant probability ofallocating to bidder i over any valuation range for which the ironed virtual valuation of bidder i is constant,then the expected revenue of A is equal to its expected ironed virtual surplus.
Fortunately, the allocation rule that maximizes ironed virtual surplus (e.g., in BSAP) would naturallyhave a constant probability of allocating to bidder i over any valuation range for which the ironed virtualvaluation of bidder i is constant. Therefore,M is simply the auction that maximizes ironed virtual surplus.For BSAP, M first computes the ironed virtual valuations of the values of all bidders. It then allocates theitem to the bidder with the highest non-negative ironed virtual valuation at a price equal to the inverse ofthe second highest one9. If there are ties, i.e., two or more bidders with maximal ironed virtual surplus thenM must break this tie consistently. (A consistent rule, for example, is to break ties in favor of bidder i overbidder j when i < j.)
Corollary 25 For any F and any incentive-compatible auction A, RM ≥ RA.
Pricings based on ironed virtual values
Since the ironed virtual valuation functions are monotone but not strictly so, their inverses are not welldefined. The standard interpretation of the inverse of an ironed virtual valuation ν is the infimum of theset of values v such that ν = φ(v). However, it will be useful for us to consider both the infimum and the
supremum of this set. So, we define←−φ −1
i (ν) = infv : φi(v) = ν, and−→φ −1
i (ν) = supv : φi(v) = ν.10
For us to relate the revenue of a pricing to the revenue of an optimal auction we must only use pricingsthat correspond to optimal auctions. In the non-regular case, optimal auctions use a consistent tie-breakingrule when two or more agents have equal ironed virtual vaulations. We extend the definition of P to thenon-regular case and define the subclass of constant-virtual-price pricings that satisfy the additional propertythat they use a consistent tie-breaking policy.
• P is the set of constant-ironed-virtual-price pricings, i.e., P = p : ∃ν ∀i φi(pi) = ν. Notice that thisdefinition coincides with our previous definition in the regular case when virtual valuations are equalto ironed virtual valuations.
• P is the set of consistent constant-ironed-virtual-price pricings, i.e., P = p : ∃ν∀i pi ∈ ←−φ −1
i (ν),−→φ −1
i (ν).This definition is well defined for both regular and non-regular distributions.
Roughly speaking we need a pricing p that satisfies (a) p ∈ Pχ=1/2 and (b) p ∈ P . Unfortunately, sucha pricing does not always exist, i.e., Pχ=1/2 may be empty. Fortunately we will show that there is a simplerounding proceedure that converts any pricing p ∈ P into a p ∈ P in a way that will not compromise ourperformance bound. To do this we need to consider the hypothetical revenue we could have gotten frompricing p if the distribution was regular. Then we need to show how to round its coordinates to get a pricingp in P in a way that allow us to convert this hypothetical revenue into actual revenue for the non-regulardistribution. We obtain a 3-approximation to RM by applying this methodology to p ∈ Pχ=1/2 if ν1/2 ≥ 0and p ∈ Pφ=0 if ν1/2 ≤ 0, i.e., to the approximately optimal pricing from Section 3.
A better accounting
Now we give an accounting trick. Rp is a complicated thing to calculate because when more than one itemis priced below the consumer’s values we must break ties in favor of the item that gives the consumer thehighest utility (i.e., with the maximum difference between the consumer’s value and the item’s price). Ouranalysis in Section 3 uses a crude lower bound for Rp in the case where two items are priced below value:the revenue is at least mini pi which, if p ∈ P , is at least φ(p) (by Fact 4). In this section we phrase ourbounds more precisely in terms of this lower bound.
9As before, the second highest non-negative ironed virtual valuation is interpreted to be 0 if there are fewer than two bidderswhose ironed virtual valuation is non-negative.
10The arrows in the notation denote “rounding down” for the infimum and “rounding up” for the supremum.
14
Definition 6 (Qp) For any p ∈ P, the revenue lower bound Qp is the expectation over v distributed as Fof the random variable Z defined as follows:
Z =
0 if vi ≤ pi for all i,
pi if vj > pj iff j = i,
φ(p) otherwise.
Fact 26 For any p ∈ P satisfying, Rp ≥ Qp.
If we replaced Rp with Qp in Section 3, all statements (specifically Lemmas 7 and 8) would remaincorrect. Thus,
Theorem 27 Under regularity either every p ∈ Pχ=1/2 or every p ∈ Pφ=0 satisfies Qp ≥ RM/3, the formerif ν1/2 ≥ 0 and the latter if ν1/2 ≤ 0.
Subsequently in this section we show that for arbitrary F and for the pricing p that we choose, we haveQp ≥ RM/3. Fact 26, then, implies our desired result.
The rounding procedure
Now we give a rounding procedure for regular distributions F that takes p ∈ P and gives p that satisfiesQp ≥ Qp.
We first note the following simple fact:
Fact 28 For p ∈ P, Qp can be calculated in polynomial time.
Definition 7 Given p ∈ P we round the coordinates to obtain p ∈ P.
1. Let p(0) = p.
2. For each i, in turn:
(a) Let ←−p and −→p satisfy ←−pi =←−φ −1
i (ν), −→pi =−→φ −1
i (ν), and ←−pj = −→pj = pj(i−1) for all j 6= i.
(b) Let p(i) = argmaxp′∈←−p ,−→p Qp′
.
3. Output p = p(n).
Lemma 29 For every regular F, virtual price ν, and pricing p ∈ Pν , there exists a p ∈ Pν that satisfiesQp ≥ Qp. The procedure in Definition 7 gives such a pricing.
The proof of Lemma 29 follows from Lemma 30, below, and a straightforward induction.
Lemma 30 For regular F, virtual price ν, and a pricing p ∈ Pν , let ←−p and −→p satisfy ←−pi =←−φ −1
i (ν),−→pi =
−→φ −1
i (ν), and ←−pj = −→pj = pj for all j 6= i. Then, maxQ←−p ,Q
−→p ≥ Qp.
Proof: Let y0 be the probability that for all indices j 6= i, vj < pj , y1 be the probability that for exactly oneindex j 6= i, vj ≥ pj, and y2 be the probability that for at least two indices j 6= i, vj ≥ pj . We can write Qp
asQp = piqiy0 + w(1 − qi)y1 + ν(qiy1 + y2)
where w is the expected revenue of the pricing p conditioned on the events that vj ≥ pj for exactly oneindex j 6= i, and vi < pi. Then,
Qp = A + B(piqi − Cqi)
where A, B and C are some constants independent of pi (but depend on pj and qj for j 6= i).
15
Consider maximizing the function Qp as a function of pi over the range [←−pi ,−→pi ]. Substituting qi =
1− Fi(pi) and differentiating with respect to pi, we get:
ddpiQp = B(C − φi(pi))fi(pi) = B(C − ν)fi(pi)
Since fi(pi) is always positive, and C and ν are constant over pi ∈ [←−pi ,−→pi ], this derivative is either always
positive or always negative over the range [←−pi ,−→pi ]. In the former case, Qp is maximized at −→pi , and in the
latter case it is maximized at ←−pi .
A regular analogy
In order to analyze the non-regular setting, we will show that there is a regular instance with propertiesvery close to that of our non-regular instance, so that some of the analysis from Section 3 carries over to thenon-regular setting. In particular, for a given non-regular distribution F there exists a regular distributionF satisfying the property that the virtual valuation function for a value distributed from F is equal tothe ironed virtual valuation function for a value distributed from F . This justifies a notational overlapwhere we let φ(·) represent the virtual valuation function with respect to F . Recall that if RF (α) denotesthe expected revenue corresponding to distribution F as a function of the probability of allocation, RF (α)denotes its concave envelope, or “ironed” revenue11. We define F in such a way that its expected revenue isRF (α) = RF (α) for all α.
Construct F from R as follows:
1. Define function g(α) = R(α)/(1 − α).
2. We show below that g(α) is strictly increasing for α ∈ [0, 1), so g−1(·) is well defined.
3. Define F (v) = g−1(v).
Lemma 31 In the above construction g(α) is strictly increasing for α ∈ [0, 1).
Proof: Upon differentiating g(α) we get:
ddαg(α) = d
dα
R(α)
1− α
=R(α)
(1 − α)2−
φ(F−1(α))
1− α
=R(α)− (1 − α)φ(F−1(α))
(1 − α)2
> 0
where the last inequality follows by observing for α < 1:
(1− α)φ(F−1(α)) <
∫ 1
α
φ(F−1(α))dα = R(α).
Note that, by definition, RF (α) = (1 − α)F−1(α) = (1− α)g(α), for all α. Therefore we have,
Fact 32 For any F and α, RF (α) = RF (α), and φF (F−1(α)) = φF (F−1(α)).
This fact immediately implies that RF (·) is concave, and therefore, φF (·) is non-decreasing.
11We drop the subscripts whenever there is no ambiguity.
16
Lemma 33 For any F , F is regular.
For all i, define Fi given RFi as above, and let F = F1× · · ·× Fn. As F is regular, the following corollaryis just a restatement of Theorem 27 for the joint distribution F.
Corollary 34 For any F, either every p in Pχ=1/2 or every p in Pφ=0 satisfies Qp
F≥ RM
F/3, the former
if ν1/2 ≥ 0 and the latter if ν1/2 ≤ 0.
The purpose of this regular analogy is to allow us to relate the quantities of interest in the non-regularcase to ones we understand in the regular case. In particular, for p ∈ P we can relate (non-regular) Qp
F to
(regular) Qp
Fand (non-regular) RMF to (regular) RM
F. The next two lemmas show that these are equalities.
Lemma 35 For any F, RMF
= RMF .
Proof: Recall that in the regular case, the revenue of Myerson’s mechanism is exactly equal to its expectedvirtual surplus. That is,
RMF
=
∫
F
max0, maxi
φFi(F−1
i (α))dα
Likewise, in the non-regular case, the revenue of Myerson’s mechanism is exactly equal to its expected ironedvirtual surplus. Therefore,
RMF =
∫
F
max0, maxi
φFi(Fi−1(α))dα
The lemma now follows from Fact 32.
Lemma 36 For any F and p ∈ P, Qp
F = Qp
F.
Proof: We first note that for p ∈ P (and indeed whenever for all i, ∃νi with pi ∈ ←−φ −1
i (νi),−→φ −1
i (νi)),R(Fi(pi)) = R(Fi(pi)). Then,
Fi(pi) = g−1i (pi) = g−1
i
(
R(Fi(pi))
1− Fi(pi)
)
= g−1i
(
R(Fi(pi))
1− Fi(pi)
)
= g−1i (gi(Fi(pi))) = Fi(pi)
Therefore, Fi(pi) = Fi(pi) for all i. Then the events, vi ≤ pi for all i, and vj > pj for some j, have equalprobabilities under F and F. The lemma now follows from the definition of Q.
Approximate pricing
We can now use the above lemmas to prove the main result of this section:
Theorem 37 For any F, there exists a pricing p in Pφ=ν1/2or Pφ=0, the former if ν1/2 ≥ 0 and the latter
if ν1/2 < 0, that satisfies Rp ≥ RM/3.
Proof: Let p be any pricing in Pχ=1/2 ⊆ Pφ=ν1/2if ν1/2 ≥ 0, or any pricing in Pφ=0 if ν1/2 < 0. Let p be
obtained by applying the rounding procedure in Definition 7 to p under regular distribution F. Then thefollowing sequence of inequalities holds.
Rp
F ≥ Qp
F Fact 26
= Qp
FLemma 36
≥ Qp
FLemma 29 applied to regular distribution F and p
≥ RMF
/3 Corollary 34
= RMF /3 Lemma 35
17
We note that Theorem 37 only gives a description of an approximately optimal pricing in the non-regular case, and not a polynomial-time approximation algorithm. We leave open the problem of designinga polynomial time algorithm for this case (in particular, a polynomial time algorithm for computing ironedvirtual valuations and their inverses), noting that for the case when each of the distributions Fi is discreteand explicitly specified, a simple algorithm for computing ironed virtual valuations has been given by Elkind(2007), and this implies a polynomial-time approximation algorithm for the non-regular case with discreteexplicit distributions.
5 Polynomial-time approximation algorithms for regular distribu-
tions
We now describe how to implement our algorithm for the regular case in the two computational modelsdescribed in Section 2. Notice that our analysis has reduced the multi-dimensional optimization problem(approximately optimal pricing) to a single dimensional optimization problem (optimal uniform virtualpricing). The remaining challenge we face will be in inverting virtual valuation functions, which we can onlydo approximately. Our final 3 + ǫ approximation is guaranteed with a 1 − o(1) probability. In Section 5.3we provide a simpler and faster algorithm that guarantees a 6 + ǫ approximation with probability 1.
5.1 The discrete case
Implementation in the discrete explicit model is straightforward. Although we have focused on continuousdistributions in Sections 3 and 4, we remark that virtual valuations and their inverses for discrete distributionscan be defined and computed in much the same way as for continuous distributions (see, e.g., Elkind (2007)).Note that the set of “valid” virtual values is finite; We extend the definition of inverses of virtual valuationsfrom the space of valid virtual values to arbitrary real numbers as follows: for any ν, let φ−1
i (ν) = minpi :φi(pi) ≥ ν, and let Pν be defined accordingly. Note that because of the discontinuity in the distributionfunction, ν1/2 does not necessarily exist in this case. Our straightforward algorithm for the discrete casecomputes virtual valuations of all possible values for each item. During this process it keeps track ofFi(φi
−1(ν)). If ν1/2 exists (that is, there is a ν with 1/2 ∈ Xν), the algorithm outputs p ∈ Pχ=1/2.Otherwise, it picks the largest ν with x < 1/2 for all x ∈ Xν and returns p ∈ Pν . The running time of eachstep of the algorithm is at most linear in n and the sizes of the supports.
If ν1/2 doesn’t exist then this process may choose p with χ(p) < 1/2. Fortunately we can fix this issueby noting that when pi = vi the consumer has zero utility for buying the item and zero utility for buyingnothing. Then, at valuation vectors where vi ≤ pi for every i, our algorithm can choose to serve or not servethe agent, and there exists an appropriate tie-breaking rule for p such that χ(p with tie-breaking) = 1/2. Byour analysis in Section 3, p with this “χ = 1/2” tie-breaking rule is a 3-approximation to the optimal pricing.However, it is easy to see that the “χ = 1/2” tie-breaking rule is not an optimal tie-breaking rule—we loserevenue by not serving the agent when pi = vi. Therefore p (without the “χ = 1/2” tie-breaking rule) isalso a 3-approximation to the optimal pricing.
5.2 The continuous case
In the continuous case, recall that we assume the distributions are specified via oracles for sampling and forevaluating the cumulative distribution function Fi and the density function fi. This case is challenging be-cause the oracle model does not allow for exact computation of inverse virtual valuations; it only allows thesequantities to be approximated. We construct an algorithm based on the analysis of Section 3. This approachis based on computing virtual valuations φi(vi) for valuations vi that are powers of (1+ ǫ), collecting a set ofcandidate pricings such that our previous analysis guarantees there is a good pricing among the collection,
18
and then sampling from the distribution to output the pricing with the best empirical performance.12 Themain technical innovation in this section is a proof that shows if we can coordinate-wise approximate apricing, we can approximate it revenue-wise as well.
5.2.1 From coordinate-wise to revenue-wise approximations
Central to our construction is a result which shows that if we can find a price point, p′, that is close (e.g.,in L∞ distance) to some desired price point, p, then we can find a new point p′′ from p′ that has revenueclose to p. This result confirms the suspicion that if a pricing can be approximated (coordinate-wise) thenso can its revenue. This result is a corollary of a lemma due to Nisan (see Balcan et al. (2005)).13
Lemma 38 For ǫ ∈ (0, 1), let p and p′ be pricings that satisfy pi′ ∈ [1 − ǫ, 1 + ǫ2 − ǫ] pi for all i. Then
Rp′
≥ (1 − 2ǫ)Rp.
Proof: Consider any valuation vector v, and let i be the index that maximizes vi − pi. In other words,when prices are given by p and a consumer has values v, the consumer buys item i. On the other hand, letj be the index that maximizes vj − pj
′. That is, when the prices are given by p′, the same consumer buysitem j instead of i. The lemma follows from the claim that
pj′ ≥ (1− 2ǫ)pi.
To prove this claim, we first observe that vi − pi ≥ vj − pj and vj − pj′ ≥ vi − pi
′. Rearranging terms andadding the two we get pi − pi
′ ≤ pj − pj′. Finally, using pi
′ ≤ (1 + ǫ2 − ǫ)pi and pj′ ≥ (1− ǫ)pj, we get
(ǫ− ǫ2)pi. ≤ ǫpj
The claim now follows by dividing by ǫ and again using the fact that pj′ ≥ (1− ǫ)pj .
Corollary 39 For ǫ ∈ (0, 1), let p and p′ be pricings that satisfy, for all i, pi′ ∈ [1− ǫ2, 1] pi. Define p′′ by
setting pi′′ = (1 + ǫ2 − ǫ)pi
′ for all i. Then Rp′′
≥ (1− 2ǫ)Rp.
Proof: Simply verify that (1− ǫ2)(1 + ǫ2 − ǫ) ≥ 1− ǫ so the conditions of Lemma 38 apply to p′′ and p.
5.2.2 Our algorithm
For a particular consumer with valuation vector v = (v1, . . . , vn), let Rp(v) be the actual revenue when pis offered to them and they choose their favorite item. Let Rp(S) be the average revenue of p when offeredto each of the consumers in a set S.
Let M = (maxi hi)/(mini ℓi). Our algorithm will run in time polynomial in n and M . For each item i,we consider the set Li of values that are powers of γ = 1/(1− ǫ2) for some ǫ > 0 in the range [ℓi, hi]. Notethat |Li| = O(logγ
hi
ℓi) = O(logγ M).
In this section, for simplicity of exposition, we assume that every virtual value has a unique inverse. Thatis, Pφ=ν is a singleton for every ν.
Definition 8 (Approximate Uniform Virtual Price Algorithm) Parameterized by constants δ and ǫin (0, 1), the approximate uniform virtual price algorithm proceeds as follows:
1. Pick the coordinate-wise largest pricing p0 ∈ Pφ=0, and evaluate χ(p0). If χ(p0) ≥ 1/2 then outputp0 and stop.
12Because of the sampling step, we can only guarantee that with high probability the price we output is good.13Though we do not discuss the details, Lemma 38 and Corollary 39 hold true even in settings where the consumers have
general combinatorial preferences.
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2. For each i and each v ∈ Li, compute φi(v) using the oracles for Fi and fi and store these in L′i.
3. For any ν, let r′(ν) = (r′1(ν), . . . , r′n(ν)) where r′i(ν) is the largest value in Li whose virtual value is atmost ν.
4. Let L′ = ν | ν ∈⋃
i Li and χ(r′(ν)) ≤ 1/2.
5. Let P = p | pi = (1 + ǫ2 − ǫ)r′i(ν′) ∀i for some ν′ ∈ L′.
6. Let S be an i.i.d. sampling of 4M2
ǫ2 log δ2|P | valuation profiles from F.
7. Output p = argmaxp′∈P Rp′
(S).
We first note that one of the pricings p ∈ P is near the approximately optimal pricing p∗ ∈ Pχ=1/2 thatour analysis of Section 3 suggests.
Lemma 40 If χ(p0) < 1/2 then there is a ν′ ∈ L′, as defined in the approximate uniform virtual pricealgorithm, and a pricing p∗ ∈ Pχ=1/2, such that for all i,
(1 − ǫ2)p∗i ≤ r′i(ν′) ≤ p∗i
Proof: First we note that r′(ν1/2) satisfies (1 − ǫ2)p∗i ≤ r′i(ν1/2) ≤ p∗i for all i. Then we show thatr′(ν1/2) = r′(ν′) for some ν′ ∈ L′.
1. For any ν and p ∈ Pφ=ν , r′(ν) satisfies (1− ǫ2)pi ≤ r′i(ν) ≤ pi for all i. This follows from the fact thatLi contains values that are powers of 1/(1− ǫ2) and φ−1
i (ν) falls between two such powers. r′i(ν) is thelower of these two powers.
2. Consider r′(ν1/2). Notice that in r′(ν1/2) each item i has a price that corresponds to a virtual valueof at most ν1/2. Let ν′ be the largest of these virtual valuations. Then r′(ν′) = r′(ν1/2). Furthermore,since ν′ ≤ ν1/2, χ(r′(ν′)) ≥ 1/2. We conclude that ν′ is in L′.
The following theorem follows directly from Theorem 14, Corollary 39, Lemma 40, and Lemma 42 (below).
Theorem 41 For any δ and ǫ in (0, 1), with probability 1−δ the approximate uniform virtual price algorithmgives a (3 + O(ǫ))-approximation to BUPP in time polynomial in n, M and 1/ǫ.
5.2.3 A sampling lemma
The final building block in our approach to approximating the pricing suggested by Section 3 is a statementthat shows that sampling from the distribution will allow us to accurately compare several pricings todetermine which has higher expected revenue. For completeness we give the full proof, though this resultfollows directly from standard approaches. The following lemma shows that for a large enough sample ofconsumers S drawn from our distribution, Rp(S) is a good approximation to Rp.
Lemma 42 For any set P of pricings with p ∈ P satisfying χ(p) ≤ 1/2 and any sample S of |S| ≥4M2
ǫ2 log δ2|P | independent draws from F,
Pr[∀p ∈ P, |Rp(S)−Rp| ≥ ǫRp] ≤ δ,
where M = maxp,i pi/ minp,i pi.
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Proof: Of course, for v drawn from F, E[Rp(v)] = Rp. We also have Rp(v) ≤ maxi pi for all v, andRp ≥ 1
2 mini pi because χ(p) ≤ 12 . Apply the Chernoff bound to get,
Pr[|Rp(S)−Rp| ≥ ǫRp] ≤ 2 exp
(
−(ǫRp)2
∑
j∈S(maxi pi/|S|)2
)
≤ 2 exp
(
−ǫ2|S|
4M2
)
≤ δ|P | .
The last step follows from the assumption that |S| ≥ 4M2
ǫ2 log δ2|P | . Now take the union bound over all p ∈ P
to prove the lemma.
5.3 A simple (6 + ǫ)-approximation
The (3 + ǫ)-approximation algorithm, though it follows from a direct approach, is rather cumbersome. Inthis section we show how to obtain a conceptually and algorithmically simple algorithm. The algorithm heregives a single pricing that is close to optimal and does not require sampling the distribution.
The algorithm is based on the observation that the Vickrey auction with appropriate non-anonymousreservation prices (henceforth, “Vickrey with reserve prices”) is a 2-approximation to the optimal single-itemauction. A similar construction to that in Section 3 can then be applied to derive a pricing that mimicsVickrey with reserve prices (and is a 3-approximation to it). This construction will only require us tocompute the inverse virtual valuation of zero, i.e., the optimal sale price, for each of the distributions. Weassume that computing this optimal sale price for each distribution Fi is possible. (Approximations can alsobe used in this step, if necessary.) Once again we assume that the distributions Fi are regular.
5.3.1 Vickrey with reserve prices
The standard interpretation of Myerson’s main result is that the optimal auction is Vickrey with an appro-priate reservation price. This, of course, is only true under regularity and when the bidders’ valuations areidentically distributed. Myerson’s main result, that maximizing welfare is equivalent to maximizing (ironed)virtual surplus, solves the single-item optimal auction problem (and more general single-parameter agentproblems) even when regularity does not hold and when the agent valuations are not identically distributed.A natural question that, to our knowledge, has not been previously answered, is measuring the extent towhich Vickrey with reservation prices approximates the optimal single-item auction when valuations are notidentically distributed. We show that, with a natural choice of reservation prices, it is a 2-approximation.
Definition 9 The Vickrey auction with reserve prices p = (p1, . . . , pn), Vp, sells an item to the bidder iwith bid bi ≥ pi that has the highest bid, if such a bidder exists. The payment that this winning bidder makesis the maximum of their reservation price pi and the bid of the highest other bidder j that bids at least theirreservation price, pj. When the bids are distributed from F we call p ∈ Pφ=0 the optimal reservation prices.
Lemma 43 For any F, the Vickrey auction with the optimal reservation prices, Vp, p ∈ Pφ=0, is a 2-approximation to the optimal single-item auction.
This lemma holds both in the regular and the non-regular case. The general proof for the irregular caseis given by Saberi and Ronen (2002). For completeness, we give a simple proof for the regular case.
Proof: We are drawing valuations v from distribution F and comparing the expected revenue of theoptimal single-item auction, M (with revenue RM), and the Vickrey auction with optimal reserve prices,Vp, p ∈ Pφ=0, (with revenue RVp). Let i be the winner of Vp and j be the winner of M (we let i = j = 0represent the case that there are no winners). Notice that i = j = 0 if and only if all virtual valuations areless than zero; and otherwise, vi ≥ vj (since Vp allocates the highest valued bidder with non-negative virtualvaluation).
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This proof uses the fact that the expected sum of payments in an auction is equal to the expected virtualsurplus (Theorem 1). Thus,
RM = E[φj(vj)]
= E[φj(vj) | i = j]Pr[i = j] + E[φj(vj) | i 6= j]Pr[i 6= j]
Both parts of this last expression can be bounded from above by RVp . For the first part,
E[φj(vj) | i = j]Pr[i = j] = E[φi(vi) | i = j]Pr[i = j]
≤ E[φi(vi)]
= RVp .
The inequality above follows from the non-negativity of virtual surplus. We bound the second part by RVp
using the fact that φj(vj) ≤ vj (Fact 4) and the fact that the payment made in Vp is at least vj as follows.
E[φj(vj) | i 6= j]Pr[i 6= j] ≤ E[vj | i 6= j]Pr[i 6= j]
≤ E[i’s payment | i 6= j]Pr[i 6= j]
≤ E[i’s payment]
= RVp .
Above, the last inequality comes from the fact that i’s payment is always non-negative. Thus,
RM ≤ 2RVp .
5.3.2 Approximating Vickrey by Pricing
This section follows almost identically to Section 3 where we showed that there is a pricing based on theMyerson’s optimal auction that approximates its revenue. Here we show that there is a pricing based on theVickrey auction with optimal reservation prices that approximates its revenue.
The pricing we consider will be p = (p1, . . . , pn) with pi = max(p0i , v), p0 ∈ Pφ=0, for v chosen such that
the probability that there is a winner is exactly 1/2 (if no such v exists then pi = p0i ). The first step of our
analysis will be to relate RVp0 to RVp which is analogous to our relating RM to RMν . Recall that Vp0 isapproximately optimal for BSAP.
Lemma 44 For any regular F, with v and p with pi = max(v, p0i ), p0 ∈ Pφ=0, satisfying χ(p) = 1/2,
Rp ≥ RVp0/3.
Proof: Consider Vp and recall that Vp allocates if and only if the pricing p does. Let x = χ(p) = χ(Vp)be the probability that neither allocates. Let qi be the probability that vi ≥ pi, i.e., qi = 1 − Fi(pi). Weproceed in steps.
1. Observe that when Vp0 allocates but Vp does not its sale price is less than v. Thus,
RVp0 ≤ RVp + vx.
2. Recall that the pricing p allocates if and only if Vp allocates and when it allocates its revenue is atleast v. Thus,
v(1− x) ≤ Rp.
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3. Lemma 8 shows that for any p,
Rp ≥ x∑
ipiqi.
4. Lemma 9 applied to Vp shows,
RVp ≤∑
ipiqi.
5. Combine the above inequalities and recall that x = 1/2. Thus,
Rp ≥ RVp0 /3.
Combining the lemma above with Lemma 13 (which shows that Rp0
≥ RM/2 when χ(p0) > 1/2) andCorollary 2 (which shows that RM ≥ RVp0 ) we obtain the main theorem of this section.
Theorem 45 For any regular F, if χ(r(0)) > 1/2 let p = p0; otherwise, let v be defined such that p withpi = max(p0
i , v) satisfies χ(p) = 1/2. Then we have
Rp ≥ RVr(0)/3.
5.3.3 Computing p
First, we assume that we are given p0 ∈ Pφ=0. (Once again, Corollary 39 implies that (1−ǫ2)-approximationsto these worsen our approximation factor by a multiplicative 1 + O(ǫ).) These are the optimal sale pricesfor each item if we were selling them alone. Now we need to compute a v such that p with pi = max(p0
i , v)the probability no items is sold (i.e., x from the above section), is (approximately) 1/2. This can be doneby binary search to arbitrary accuracy. Notice that if we do not have x = 1/2 exactly, plugging x = 1/2− ǫinto the above theorem gives us Rp ≥ RVp0/(3 + ǫ) (and using Lemma 43, Rp ≥ RM/(6 + ǫ)).
Theorem 46 For regular F, there is a polynomial time algorithm that gives a (6 + ǫ)-approximation toBUPP.
6 Conclusions
Several interesting questions related to BUPP still remain open:
• Is the Bayesian unit-demand pricing problem with independently distributed values NP-hard to solveoptimally? There is some evidence that this problem is indeed hard. For example, one can constructtwo-item instances with extremely simple distributions (e.g., a uniform distribution over some range),where the optimal price is irrational.
• Is our characterization tight? Can one construct an example where the revenue of Myerson’s auctionis indeed three times the revenue of the optimal pricing? It is worth noting that there is a simpleexample in which the revenue of the pricing defined by our virtual valuation technique falls short ofthe optimal pricing by a factor of nearly 2, even in the i.i.d. case. (See Lemma 21 in Section 3.2.)
• Extending this work to accommodate combinatorial consumers seems tricky. An optimal pricing inthat case may offer bundles at prices higher or lower than the sum of the prices of individual items inthe bundle.
• Finally, a more general selling mechanism in the unit-demand case may offer lotteries to consumers.A lottery is a distribution over single items, sold at a price (typically) lower than the prices for theindividual items. Thanassoulis (2004) was the first to note that lotteries can in general obtain higherrevenue than item pricings. In fact when values are correlated, the revenue of the optimal single-item
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pricing can be an exponential factor smaller than the revenue of the optimal collection of lotteries. Therevenue of the optimal collection of lotteries is not always bounded above by the revenue of Myerson’sauction, so the techniques in this paper do not imply an approximation to the optimal lottery system.
Finally, the most important question related to this work is whether our techniques can be extended toobtain concise descriptions for approximately optimal solutions to more general multi-parameter mechanismdesign problems.
Acknowledgments
We thank Nina Balcan for several useful discussions.
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